guide:158667739d

[math] \newcommand{\indep}[0]{\ensuremath{\perp\!\!\!\perp}} \newcommand{\dpartial}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\abs}[1]{\left| #1 \right|} \newcommand\autoop{\left(} \newcommand\autocp{\right)} \newcommand\autoob{\left[} \newcommand\autocb{\right]} \newcommand{\vecbr}[1]{\langle #1 \rangle} \newcommand{\ui}{\hat{\imath}} \newcommand{\uj}{\hat{\jmath}} \newcommand{\uk}{\hat{k}} \newcommand{\V}{\vec{V}} \newcommand{\half}[1]{\frac{#1}{2}} \newcommand{\recip}[1]{\frac{1}{#1}} \newcommand{\invsqrt}[1]{\recip{\sqrt{#1}}} \newcommand{\halfpi}{\half{\pi}} \newcommand{\windbar}[2]{\Big|_{#1}^{#2}} \newcommand{\rightinfwindbar}[0]{\Big|_{0}^\infty} \newcommand{\leftinfwindbar}[0]{\Big|_{-\infty}^0} \newcommand{\state}[1]{\large\protect\textcircled{\textbf{\small#1}}} \newcommand{\shrule}{\\ \centerline{\rule{13cm}{0.4pt}}} \newcommand{\tbra}[1]{$\bra{#1}$} \newcommand{\tket}[1]{$\ket{#1}$} \newcommand{\tbraket}[2]{$\braket{1}{2}$} \newcommand{\infint}[0]{\int_{-\infty}^{\infty}} \newcommand{\rightinfint}[0]{\int_0^\infty} \newcommand{\leftinfint}[0]{\int_{-\infty}^0} \newcommand{\wavefuncint}[1]{\infint|#1|^2} \newcommand{\ham}[0]{\hat{H}} \newcommand{\mathds}{\mathbb}[/math]

\label{sec:ate} In this particular case, we are interested in a number of causal quantities. The most basic and perhaps most important one is whether the treament is effective (i.e., results in a positive outcome) generally. This corresponds to checking whether [math]\mathbb{E}\left[y | \mathrm{do}(a=1)\right] \gt \mathbb{E}\left[y | \mathrm{do}(a=0)\right][/math], or equivalently computing

[[math]] \begin{align} \mathrm{ATE} = \mathbb{E}\left[y | \mathrm{do}(a=1)\right] - \mathbb{E}\left[y | \mathrm{do}(a=0)\right], \end{align} [[/math]]

where

[[math]] \begin{align} \mathbb{E}\left[y | \mathrm{do}(a=\hat{a})\right] &= \sum_{y} y p(y | \mathrm{do}(a=\hat{a})) \\ &= \sum_{y} y \sum_{x} p(x) p(y|\hat{a}, x) = \sum_{y} y \mathbb{E}_{x \sim p(x)} \left[ p(y|\hat{a}, x) \right]. \end{align} [[/math]]

In words, we average the effect of [math]\hat{a}[/math] on [math]y[/math] over the covariate distribution but the choice of [math]\hat{a}[/math] should not depend on [math]x[/math]. Then, we use this interventional distribution [math]p(y | \mathrm{do}(a=\hat{a}))[/math] to compute the average outcome. We then look at the difference in the average outcome between the treatment and not (placebo), to which we refer as the average treatment effect (ATE). It is natural to extend ATE such that we do not marginalize the entire covariate [math]x[/math], but fix some part to a particular value. For instance, we might want to compute ATE but only among people in their twenties. Let us rewrite the covariate [math]x[/math] as a concatenation of [math]x[/math] and [math]x'[/math], where [math]x'[/math] is what we want to condition ATE on. That is, instead of [math]p(y | \mathrm{do}(a=\hat{a}))[/math], we are interested in [math]p(y | \mathrm{do}(a=\hat{a}), x'=\hat{x}')[/math]. This corresponds to first modifying [math]G[/math] into \begin{center} \begin{tikzpicture}

  \node[latent] (a) {[math]a[/math]};   \node[latent, right=1cm of a] (y) {[math]y[/math]};   \node[latent, above=0.5cm of a, xshift=0.5cm] (x) {[math]x[/math]};   \node[obs, right=0.5cm of x] (xc) {[math]\hat{x}'[/math]};   
  \edge{x}{a}; 
\edge{x}{y};
\edge{xc}{x}; 
\edge{xc}{a}; 
\edge{xc}{y};
\edge{a}{y};

\end{tikzpicture} \end{center} and then into \begin{center} \begin{tikzpicture}

  \node[latent] (a) {[math]a[/math]};   \node[latent, right=1cm of a] (y) {[math]y[/math]};   \node[latent, above=0.5cm of a, xshift=0.5cm] (x) {[math]x[/math]};   \node[obs, right=0.5cm of x] (xc) {[math]\hat{x}'[/math]};   
  \edge{x}{y};
\edge{xc}{x}; 
\edge{xc}{y};
\edge{a}{y};

\end{tikzpicture} \end{center} We then get the following conditional average treatment effect (CATE):

[[math]] \begin{align} \mathrm{CATE} = \mathbb{E}\left[y | \mathrm{do}(a=1), x'=\hat{x}'\right] - \mathbb{E}\left[y | \mathrm{do}(a=0), x'=\hat{x}'\right], \end{align} [[/math]]

where

[[math]] \begin{align} \mathbb{E}\left[y | \mathrm{do}(a=\hat{a}), x'=\hat{x}'\right] &= \sum_{y} y p(y | \mathrm{do}(a=\hat{a}), x'=\hat{x'}) \\ &= \sum_{y} y \sum_{x} p(x|x') p(y|\hat{a}, x'=\hat{x'}, x) \\ &= \sum_{y} y \mathbb{E}_{x \sim p(x|x')} \left[ p(y|\hat{a}, x'=\hat{x'}, x) \right]. \end{align} [[/math]]

You can see that this is really nothing but ATE conditioned on [math]x'=\hat{x}'[/math]. From these two quantities of interest above, we see that the core question is whether and how to compute the interventional probability of the outcome [math]y[/math] given the intervention on the action [math]a[/math] conditioned on the context [math]x'[/math]. Once we can compute this quantity, we can computer various target quantities under this distribution. We thus do not go deeper into other widely used causal quantities in this course but largely stick to ATE/CATE.

General references

Cho, Kyunghyun (2024). "A Brief Introduction to Causal Inference in Machine Learning". arXiv:2405.08793 [cs.LG].